Using the reproducing property of the Szegö kernel of \(H_{\mathcal {E}}^2(\) , one can easily show that \(M_z \in \mathcal {B}(H_{\mathcal {E}}^2()\)  is a pure contraction. In fact, the remarkable Sz-Nagy-Foias model (see [1]}) for a contraction proves that a pure contraction upto unitary equivalence is the compression operator \(P_{\mathcal {Q}} M_z|_{\mathcal {Q}}\)  where, \(\mathcal {Q}\)  is a \(M_z^*\)  invariant closed subspace of some vector-valued Hardy space. An important characterization that follows from the above model shows that: a contraction \(T\)  on \(\mathcal {H}\)  is pure if and only if its characteristic function \(\Theta _T(z)\)  is a inner function (see [1]}). Although this is a complete characterization, determining the behaviour of \(\Theta _T\)  is a challenging problem in itself and, thus, it is seldom used for checking pure contractions. Hence, it is a natural question to find more examples of pure contractions and more importantly, to find a tractable criterion. Before stating our main question, let us establish the following convention.
